Note that \({\rm sn}(z,m)\)  is meromorphic in both \(z\)  and \(m\) , and doubly periodic in \(z\) : \({\rm sn}(z,m)={\rm sn}(z+n\omega +n^{\prime }\omega ^{\prime },m)\) , for \(n,n^{\prime }\in \mathbb {Z}\) .
The two periods \(\omega \)  and \(\omega ^{\prime }\)  point in different directions in the complex plane (i.e. \(\omega /\omega ^{\prime }\notin \mathbb {R}\) ) and are given by \(\omega =4K\)  and \(\omega ^{\prime }=2i K^{\prime }\) , where \(K\equiv K(m)\) , \(K^{\prime }\equiv K(1-m)\) , with \(K(m)\)  the complete elliptic integral of the first kind [1]}. Thus, we can think of \({\rm sn}(z,m)\)  as a periodic decoration or tiling of the complex \(z\)  plane, whose fundamental parallelogram has edges \(\omega \)  and \(\omega ^{\prime }\) . Alternatively, we can think of \({\rm sn}(z,m)\)  as a function over the toroidal Riemann surface formed by identifying the opposite edges of this fundamental parallelogram. The zeros of \({\rm sn}(z,m)\)  are at \(z=2nK+2n^{\prime }iK^{\prime }\)  and the poles are at \(z=2nK+(2n^{\prime }+1)iK^{\prime }\) , with \(n,n^{\prime }\in \mathbb {Z}\) .
